Single-molecule Forster resonance energy transfer (FRET) between fluorescent donor and acceptor labels attached to a protein or nucleic acid can be used to probe a molecules structure, dynamics and function. In these experiments, a molecule is either immobilized on a surface or diffuses through a spot illuminated by a laser, and the donor is excited. The donor can emit a photon or transfer the excitation to an acceptor which then can emit a photon of a different color. The rate of transfer depends on (interdye distance)-6 and this is why there is information about conformational dynamics (FRET is the optical analog of the NOE in NMR that is used in structure determination). The output of these experiments is a photon trajectory (the color of the photons emitted by the donor differ from those emitted by the acceptor). The observed sequence of photons can be binned, and a histogram of the FRET efficiencies for each bin, defined as the fraction of the photons emitted from the acceptor, can be constructed. The shape of the histogram depends on the conformational states of the molecule and their interconversion rates. Alternatively, photon sequences can be analyzed without binning using likelihood-based methods. We have developed a rigorous theoretical framework for the analysis of such single-molecule FRET experiments. The theory describes how statistics of photons are influenced by conformational dynamics, diffusion of the molecule through the laser spot, shot noise, dye photophysics, etc. Various aspects of the theory have been summarized in a comprehensive review (see reference 1). The theory has been used in the analysis of single-molecule experiments performed in the laboratory of Dr. W. A. Eaton (LCP, NIDDK). When a single molecule is excited by a train of laser pulses, it is not only possible to detect the colors and arrival times of the emitted photons, but also the time interval between the laser pulse and the photon. This so-called delay time is related to the fluorescence lifetime of the fluorophore. The fluorescence lifetime depends on the rate of energy transfer and hence decreases as the donor and acceptor come closer together. In 2, we have generalized our previous work on FRET efficiency histograms to include delay times. Our main theoretical contribution was to derive an exact expression for the joint distribution of the numbers of donor and acceptor photons and donor lifetimes in a bin that treats the influence of conformational dynamics on all time scales. Perhaps the most interesting finding is that the connectivity of the underlying conformational states can be determined directly by simple visual inspection of the projection of the experimental joint distribution on the efficiency-lifetime plane. In a complimentary approach, the whole photon trajectory is analyzed by determining how well a model describes it. This is done by maximizing the appropriate likelihood function with respect to the parameters of the model of conformational dynamics. In 3, our previous likelihood-based analysis was extended to a class of experiments in which only the number of photons in consecutive time intervals is recorded. This allows one to improve the likelihood function that is traditionally used in Hidden Markov Models. The rate with which an enzyme with a single catalytic site converts a substrate into product, depends on the substrate concentration in a hyperbolic way, as described by the well-known Michaelis- Menten equation. In classical enzyme kinetics, deviations from such hyperbolic behavior is taken to be a hallmark of cooperativity (1.e. the enzyme has several interacting binding sites). In 4, we show that deviations from the predictions of the Michaelis-Menten equation can occur even if there is there is a single binding site as a result of the diffusive nature of substate binding. The basic idea behind out mathematical theory, is that at high substate concentrations, diffusion is unimportant because some substrate is always close to the binding site. At low substate concentration, they are far apart and the substrate has to difuusive a significant dististance to reach the binding. Thus the binding rates are different different in the two cases, resulting in a more complex dependence of the turnover rate on substrate concentration. Our theoretical predictions avaits experimental confirmation.